A recurring challenge in the application of redistricting simulation algorithms lies in extracting useful summaries and comparisons from a large ensemble of districting plans. Researchers often compute summary statistics for each district in a plan, and then study their distribution across the plans in the ensemble. This approach discards rich geographic information that is inherent in districting plans. We introduce the projective average, an operation that projects a district-level summary statistic back to the underlying geography and then averages this statistic across plans in the ensemble. Compared to traditional district-level summaries, projective averages are a powerful tool for geographically granular, sub-district analysis of districting plans along a variety of dimensions. However, care must be taken to account for variation within redistricting ensembles, to avoid misleading conclusions. We propose and validate a multiple-testing procedure to control the probability of incorrectly identifying outlier plans or regions when using projective averages.

翻译：在选区划分模拟算法的应用中，一个反复出现的挑战在于如何从大量选区划分方案的集成中提取有用的汇总信息和进行有效比较。研究者通常计算每个方案中各选区的汇总统计量，进而研究这些统计量在整个集成方案中的分布。然而，这种方法忽略了选区划分方案中固有的丰富地理信息。我们引入投影平均这一操作，它可将选区层面的汇总统计量投影至底层地理空间，随后对集成方案中该统计量取平均值。与传统选区层面的汇总方法相比，投影平均是沿多种维度对选区划分方案进行地理粒度精细化的亚选区分析的强效工具。但需注意控制选区划分集成内部的变异性，以避免得出误导性结论。我们提出并验证了一种多重检验程序，可在使用投影平均时有效控制错误识别异常方案或异常区域的概率。